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<?xml version="1.0" encoding="UTF-8"?>
<!--
* Scilab ( http://www.scilab.org/ ) - This file is part of Scilab
* Copyright (C) 2006-2008 - INRIA
*
* This file must be used under the terms of the CeCILL.
* This source file is licensed as described in the file COPYING, which
* you should have received as part of this distribution. The terms
* are also available at
* http://www.cecill.info/licences/Licence_CeCILL_V2.1-en.txt
*
-->
<refentry xmlns="http://docbook.org/ns/docbook" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:svg="http://www.w3.org/2000/svg" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:db="http://docbook.org/ns/docbook" xmlns:scilab="http://www.scilab.org" xml:lang="en" xml:id="sylm">
<refnamediv>
<refname>sylm</refname>
<refpurpose>Sylvester matrix</refpurpose>
</refnamediv>
<refsynopsisdiv>
<title>Calling Sequence</title>
<synopsis>[S]=sylm(a,b)</synopsis>
</refsynopsisdiv>
<refsection>
<title>Arguments</title>
<variablelist>
<varlistentry>
<term>a,b</term>
<listitem>
<para>two polynomials</para>
</listitem>
</varlistentry>
<varlistentry>
<term>S</term>
<listitem>
<para>matrix</para>
</listitem>
</varlistentry>
</variablelist>
</refsection>
<refsection>
<title>Description</title>
<para>
<literal>sylm(a,b)</literal> gives the Sylvester matrix associated to polynomials
<literal>a</literal> and <literal>b</literal>, i.e. the matrix <literal>S</literal> such that:
</para>
<para>
<literal>coeff( a*x + b*y )' = S * [coeff(x)';coeff(y)']</literal>.
</para>
<para>
Dimension of <literal>S</literal> is equal to <literal>degree(a)+degree(b)</literal>.
</para>
<para>
If <literal>a</literal> and <literal>b</literal> are coprime polynomials then
</para>
<para>
<literal>rank(sylm(a,b))=degree(a)+degree(b))</literal> and the instructions
</para>
<programlisting role="no-scilab-exec"><![CDATA[
u = sylm(a,b) \ eye(na+nb,1)
x = poly(u(1:nb),'z','coeff')
y = poly(u(nb+1:na+nb),'z','coeff')
]]></programlisting>
<para>
compute Bezout factors <literal>x</literal> and <literal>y</literal> of minimal degree
such that <literal>a*x+b*y = 1</literal>
</para>
</refsection>
<refsection>
<title>Examples</title>
<programlisting role="example"><![CDATA[
x = poly(0,"x")
y = poly ([1, 2, 3], "x","coeff");
sylm(x,y)
]]></programlisting>
</refsection>
</refentry>
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